Meeting Details

In this talk we survey normal bases and normal elements in finite fields. These concepts were defined, and their existence proved, 150 years ago. However, due to their many recent applications, they have been vastly studied in the last 25 years.
Let q be a prime power. An element \alpha in a finite field
F_{q^n} is called "normal" if N={\alpha, \alpha^q, ..., \alpha^{q^{n-1}}} is a basis of F_{q^n} over F_q. In this case, the basis N is called a "normal basis" of F_{q^n} over F_q.
First we briefly give an account of basic properties and results on normal elements including existence and number of normal elements.
Then we focus on how to operate with normal basis. As Hensel noted, in a normal basis q-th powers are for free. This can be exploited to have fast exponentiation algorithms. As a consequence, normal elements are important in cryptographic applications where exponentiation and discrete logarithm
computations are employed.
Next we discuss how to find normal elements. It turns out that not all normal elements behave in the same way, the so called "optimal normal elements" being preferable for most computations with normal elements. These special
elements are directly related to Gauss periods in finite fields and have been characterized by Gao and Lenstra. Unfortunately, optimal normal elements only exists for some extension fields. This makes the study of "low complexity" normal elements relevant. We comment on several old and new results to produce low complexity normal elements. We conclude giving some open problems.